3.18.4 Tensor Dual Basis Notation

Tensor Dual Basis Notation defines dual vectors using a basis, key for tensor algebra in math and physics.

Tensor Dual Basis Notation is the system of symbols used to denote the basis of a dual vector space that is constructed from a chosen basis of the original vector space, together with the index conventions that make this construction compatible with tensor component notation. It provides the standard means of writing covector components, of expressing the Kronecker delta relation that characterizes the dual basis, and of establishing the raised-index convention used throughout tensor algebra to distinguish covariant from contravariant quantities.

Construction of the Dual Basis

Starting Basis and Dual Space

Given a finite-dimensional vector space V with basis e_1, ..., e_n, the dual space V* consists of all linear functionals from V to the base field. The dual basis notation assigns to this basis a corresponding set of functionals e^1, ..., e^n in V*, distinguished from the primal basis vectors by superscript indices rather than subscript indices. This typographic convention, superscripts for dual basis elements and subscripts for primal basis elements, is the first and most fundamental rule of dual basis notation.

Defining Relation

Each dual basis covector e^i is defined as the unique linear functional satisfying the Kronecker delta pairing against every primal basis vector.

e^{i} (e_{j}) = δ_{j}^{i}

where the Kronecker delta equals one when i equals j and zero otherwise. This defining relation guarantees that e^i extracts precisely the i-th coordinate of any vector expressed in the primal basis, since applying e^i to a general vector v with components v^j yields v^i by linearity and the delta relation.

Index Placement Conventions

Upper and Lower Indices

Dual basis notation is inseparable from the broader convention of upper and lower indices in tensor algebra. Components of vectors, being coefficients in the primal basis e_j, carry upper indices, written v^j, while components of covectors, being coefficients in the dual basis e^i, carry lower indices, written alpha_i. A covector alpha is expanded in the dual basis as follows.

α = \sum_{i = 1}^{n} α_{i} e^{i}

Einstein Summation with Dual Basis

Under the Einstein summation convention, repeated indices appearing once as a superscript and once as a subscript are automatically summed, so the expansion above is written compactly as alpha_i e^i. This convention only functions correctly because dual basis notation places the basis covector index as a superscript and the component index as a subscript, ensuring that every summation in the theory pairs one upper and one lower index, which is precisely the algebraic signature of a scalar invariant under change of basis.

Change of Basis Behavior

Transformation of the Primal Basis

If a new primal basis is related to the old one by a transformation matrix A, so that the new basis vectors are linear combinations of the old ones, then vector components transform contravariantly, meaning with the inverse of A.

{\tilde{e}}_{i} = \sum_{j = 1}^{n} A_{i}^{j} e_{j}

Transformation of the Dual Basis

The dual basis, by contrast, transforms with the inverse-transpose of A, which is the defining reason it is called contragredient to the primal basis. If B denotes the inverse of A, the new dual basis vectors satisfy the following relation.

{\tilde{e}}^{i} = \sum_{j = 1}^{n} B_{j}^{i} e^{j}

This inverse relationship is what preserves the Kronecker delta pairing between the new primal basis and the new dual basis, and it is the reason the notation labels dual basis indices with superscripts: the transformation rule for these superscripted quantities mirrors the transformation rule for vector components, not for the primal basis vectors themselves, which is the origin of the terms covariant and contravariant.

Extension to Tensor Product Bases

Bases for Tensor Spaces of Higher Rank

Dual basis notation extends directly to the construction of bases for tensor product spaces. A rank-two covariant tensor lives in V* tensor V*, and its natural basis consists of tensor products of dual basis covectors, written e^i tensor e^j, with two independent superscript indices. A mixed tensor of type (1,1), living in V tensor V*, uses a basis element e_i tensor e^j carrying one lower and one upper index, matching the index structure of its components T^i_j.

Component Extraction via Pairing

For a general tensor T of type (p, q), its components relative to these product bases are recovered by pairing T against the appropriate combination of dual basis covectors and primal basis vectors.

T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = T (e^{i_{1}}, \dots, e^{i_{p}}, e_{j_{1}}, \dots, e_{j_{q}})

This formula shows why the dual basis is indispensable beyond the covector case: it supplies the exact set of test objects, the e^i, needed to probe and extract every contravariant index of an arbitrary tensor, just as the primal basis vectors supply the test objects needed to probe covariant indices.

Practical Role in Pullback Formulas

Coordinate Expression of the Pullback

When a linear map f from V to W is expressed in coordinates relative to chosen primal bases of V and W, the pullback of a dual basis covector of W* can be expanded in the dual basis of V* using the same matrix that represents f, transposed. If f has matrix representation with entries f^a_i in the given bases, the pullback of the dual basis covector e^a of W* is expressed as follows.

f^{*} e^{a} = \sum_{i = 1}^{n} f_{i}^{a} e^{i}

This identity is the working formula behind matrix transposition in linear algebra: the matrix of the pullback map on dual bases is exactly the transpose of the matrix of the original map on primal bases, and dual basis notation is what allows this fact to be stated as a clean index identity rather than as an ad hoc rule about swapping rows and columns.